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C₄×#1629 #209661

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂×D₄²	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2,3,4);(5,6,8,11)(7,9,12,10)(13,14,17,20)(15,18,16,19);(6,9)(7,12)(10,11)(14,18)(15,16)(19,20);(14,19)(15,16)(18,20)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#18524, AGL(1,𝔽₅)×#1629
maximal normal subgroup of	C₄×#18524
maximal quotient of	C₄×#12364, C₄×#8886, C₄×#12366, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116857);